Subgroups — C₂×#19134171 #25079171

Order 11008.

694 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583250	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583190	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583180	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582978	C₂	1	1	yes	maximal, characteristic
5	#19134171	C₂	1	8	yes	maximal, direct product factor
6	C₂×#14582647	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14582644	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10621872	—	2	1	no	—
10	C₂×#13063325	C₂²	1	1	yes	characteristic
11	C₂×#13063325	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
13	#14583250	—	2	2	no	—
14	#14583250	C₂²	1	4	yes	complemented
15	#14583190	—	2	2	no	—
16	#14583190	C₂²	1	4	yes	—
17	#14583180	—	2	2	no	—
18	#14583180	C₂²	1	4	yes	—
19	#14582978	—	2	2	no	—
20	#14582978	C₂²	1	4	yes	—
21	C₂×#13063259	—	2	1	no	—
22	C₂×#13063257	—	2	1	no	—
23	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
24	C₂×#13063250	—	2	1	no	—
25	C₂×#13063250	—	2	1	no	—
26	C₂×#13063247	—	2	1	no	—
27	C₂×#13063247	—	2	1	no	—
28	C₂×(C₂×D₄₃)⋊C₈	—	2	1	no	—
29	#14582647	C₂²	1	4	yes	complemented
30	#14582644	C₂²	1	4	yes	complemented
31	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
34	C₄₃×#334	C₂²	1	4	yes	complemented
35	C₂²×Dic₈₆	—	2	1	no	—
36	C₂²×Dic₈₆	—	2	1	no	—
37	C₂²×D₁₇₂	D₄	1	1	yes	characteristic
38	C₂²×C₄×D₄₃	—	2	1	no	—
39	C₂×#10621872	—	2	4	no	—
40	C₂×#10621872	—	4	4	no	—
41	C₂²×C₄₃⋊C₈	—	2	1	no	—
42	#13063325	—	2	1	no	—
43	#13063325	—	2	1	no	—
44	#13063325	—	2	1	no	—
45	#13063325	—	2	1	no	—
46	#13063325	—	2	1	no	—
47	#13063325	—	2	1	no	—
48	#13063325	C₂³	1	2	yes	—
49	#13063325	C₂³	1	2	yes	—
50	C₄×D₁₇₂	—	2	1	no	—
51	C₄×D₁₇₂	—	2	1	no	—
52	C₄×D₁₇₂	D₄	1	2	yes	—
53	C₄×D₁₇₂	C₂³	1	2	yes	—
54	C₂×C₄⋊Dic₄₃	—	2	1	no	—
55	C₂×C₄⋊Dic₄₃	—	2	1	no	—
56	C₂×C₄⋊Dic₄₃	—	2	1	no	—
57	C₂×C₄⋊Dic₄₃	—	2	1	no	—
58	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
59	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
60	#13063259	—	2	4	no	—
61	#13063257	—	2	4	no	—
62	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
63	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
64	#13063250	—	2	4	no	—
65	#13063250	—	2	4	no	—
66	#13063247	—	2	4	no	—
67	#13063247	—	2	4	no	—
68	(C₂×D₄₃)⋊C₈	—	2	4	no	—
69	C₂²×C₈×C₄₃	—	2	1	no	—
70	C₂²×C₈×C₄₃	—	2	1	no	—
71	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
72	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₄₃	—	2	1	no	—
75	C₄⋊C₈×C₄₃	—	2	1	no	—
76	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
77	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
78	C₄×C₈×C₄₃	D₄	1	2	yes	—
79	C₄×C₈×C₄₃	C₂³	1	2	yes	—
80	C₂³×D₄₃	—	2	1	no	—
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
84	C₂²×Dic₄₃	—	2	1	no	—
85	C₂²×Dic₄₃	—	4	1	no	—
86	C₂²×Dic₄₃	—	4	1	no	—
87	C₂×D₁₇₂	—	2	1	no	—
88	C₂×D₁₇₂	—	2	2	no	—
89	C₂×D₁₇₂	—	2	2	no	—
90	C₂×D₁₇₂	C₂×D₄	1	2	yes	—
91	C₂×C₄×D₄₃	—	2	2	no	—
92	C₂×C₄×D₄₃	—	2	2	no	—
93	C₂×C₄×D₄₃	—	4	2	no	—
94	C₂×C₄×D₄₃	—	4	2	no	—
95	C₂×Dic₈₆	—	2	2	no	—
96	C₂×Dic₈₆	—	2	2	no	—
97	C₂×Dic₈₆	—	4	2	no	—
98	C₂×Dic₈₆	—	4	2	no	—
99	C₂×Dic₈₆	—	4	1	no	—
100	C₂×Dic₈₆	—	4	1	no	—
101	C₂×Dic₈₆	—	8	2	no	—
102	C₂×C₈×C₄₃	—	2	2	no	—
103	C₂×C₈×C₄₃	—	2	2	no	—
104	C₂×C₈×C₄₃	—	2	2	no	—
105	C₂×C₈×C₄₃	—	2	2	no	—
106	C₂×C₈×C₄₃	—	4	2	no	—
107	C₂×C₈×C₄₃	—	4	2	no	—
108	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
110	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₄₃	#41	1	1	yes	characteristic
113	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
114	(C₂×D₄₃)⋊C₄	—	2	2	no	—
115	(C₂×D₄₃)⋊C₄	—	2	2	no	—
116	C₄⋊Dic₄₃	—	2	1	no	—
117	C₄⋊Dic₄₃	—	2	2	no	—
118	C₄⋊Dic₄₃	—	2	2	no	—
119	C₄⋊Dic₄₃	—	2	2	no	—
120	C₄⋊Dic₄₃	—	2	2	no	—
121	C₄⋊Dic₄₃	—	4	1	no	—
122	C₄⋊Dic₄₃	—	4	1	no	—
123	C₄⋊Dic₄₃	—	4	1	no	—
124	C₄⋊Dic₄₃	—	4	1	no	—
125	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
126	C₂×C₄₃⋊C₈	—	2	2	no	—
127	C₂×C₄₃⋊C₈	—	4	2	no	—
128	#10621872	—	4	8	no	—
129	C₂³×C₄₃	#121	1	1	yes	characteristic
130	C₂²×D₄₃	—	2	2	no	—
131	C₂²×D₄₃	—	4	2	no	—
132	C₂²×D₄₃	—	4	2	no	—
133	C₂²×D₄₃	—	4	2	no	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
137	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#135	1	2	yes	—
139	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
140	C₂×C₄×C₄₃	#137	1	2	yes	—
141	C₂×C₄×C₄₃	#137	1	2	yes	—
142	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
145	C₂×Dic₄₃	—	2	2	no	—
146	C₂×Dic₄₃	—	4	2	no	—
147	C₂×Dic₄₃	—	4	2	no	—
148	C₂×Dic₄₃	—	4	2	no	—
149	C₂×Dic₄₃	—	8	2	no	—
150	C₂×Dic₄₃	—	8	2	no	—
151	D₁₇₂	—	2	2	no	—
152	D₁₇₂	—	2	2	no	—
153	D₁₇₂	—	4	2	no	—
154	C₄×D₄₃	—	4	2	no	—
155	C₄×D₄₃	—	4	2	no	—
156	C₄×D₄₃	—	4	4	no	—
157	Dic₈₆	—	4	2	no	—
158	Dic₈₆	—	4	2	no	—
159	Dic₈₆	—	8	2	no	—
160	Dic₈₆	—	8	2	no	—
161	Dic₈₆	—	8	2	no	—
162	C₈×C₄₃	—	2	2	no	—
163	C₈×C₄₃	—	2	2	no	—
164	C₈×C₄₃	—	4	2	no	—
165	C₈×C₄₃	—	4	2	no	—
166	C₄₃⋊C₈	—	4	2	no	—
167	C₂×#1703	—	43	1	no	maximal, Sylow
168	C₂²×C₄₃	#463	1	2	yes	—
169	C₂²×C₄₃	#470	1	2	yes	—
170	C₂²×C₄₃	#471	1	2	yes	—
171	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
172	C₂×D₄₃	—	4	2	no	—
173	C₂×D₄₃	—	4	2	no	—
174	C₂×D₄₃	—	4	2	no	—
175	C₂×D₄₃	—	8	4	no	—
176	C₄×C₄₃	—	2	1	no	—
177	C₄×C₄₃	—	2	1	no	—
178	C₄×C₄₃	#463	1	2	yes	—
179	C₄×C₄₃	#470	1	2	yes	—
180	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
181	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
183	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
184	Dic₄₃	—	4	2	no	—
185	Dic₄₃	—	8	2	no	—
186	Dic₄₃	—	8	2	no	—
187	C₂×#487	—	43	1	no	—
188	C₂×#476	—	43	1	no	—
189	C₂×#474	—	43	1	no	—
190	C₂×#436	—	43	1	no	—
191	#1703	—	43	8	no	—
192	C₂×#334	D₄₃	1	1	yes	characteristic, complemented
193	C₂×#333	—	43	1	no	—
194	C₂×#332	—	43	1	no	—
195	C₂×C₄₃	#1703	1	4	yes	complemented
196	C₂×C₄₃	C₂×#463	1	1	yes	characteristic
197	C₂×C₄₃	C₂×#470	1	1	yes	characteristic
198	C₂×C₄₃	C₂×#471	1	1	yes	characteristic
199	D₄₃	—	8	2	no	—
200	C₂²×M₁₆	—	86	1	no	—
201	C₂×C₄⋊Q₈	—	43	1	no	—
202	C₂×C₄⋊Q₈	—	43	1	no	—
203	C₂×C₄×D₄	—	43	1	no	—
204	#487	—	43	4	no	—
205	#487	—	86	2	no	—
206	#476	—	43	4	no	—
207	#476	—	86	2	no	—
208	#474	—	43	4	no	—
209	#474	—	86	2	no	—
210	#436	—	43	4	no	—
211	#436	—	86	2	no	—
212	C₂×#107	—	86	1	no	—
213	C₂×#106	—	86	1	no	—
214	C₂×C₄⋊C₈	—	43	1	no	—
215	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
216	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
217	C₂×#103	—	86	1	no	—
218	C₂×#103	—	86	1	no	—
219	C₂×#102	—	86	1	no	—
220	C₂×#102	—	86	1	no	—
221	C₂×C₂²⋊C₈	—	86	1	no	—
222	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
223	#334	C₂×D₄₃	1	4	yes	complemented
224	#333	—	43	4	no	—
225	#332	—	43	4	no	—
226	C₄₃	C₂×#1703	1	1	yes	characteristic, complemented, Sylow
227	C₂²×Q₈	—	86	1	no	—
228	C₂²×Q₈	—	86	1	no	—
229	C₂²×D₄	—	43	1	no	—
230	C₂³×C₄	—	86	1	no	—
231	C₂×M₁₆	—	86	4	no	—
232	C₂×M₁₆	—	172	4	no	—
233	C₂²×C₈	—	2	1	no	—
234	C₂²×C₈	—	2	1	no	—
235	C₂²×C₈	—	86	1	no	—
236	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
237	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
238	C₄⋊Q₈	—	43	2	no	—
239	C₄⋊Q₈	—	43	2	no	—
240	C₄⋊Q₈	—	86	1	no	—
241	C₄⋊Q₈	—	86	1	no	—
242	C₄⋊Q₈	—	86	1	no	—
243	C₄⋊Q₈	—	86	1	no	—
244	C₄⋊Q₈	—	86	1	no	—
245	C₄⋊Q₈	—	86	1	no	—
246	C₄×D₄	—	43	2	no	—
247	C₄×D₄	—	43	2	no	—
248	C₄×D₄	—	86	1	no	—
249	C₄×D₄	—	86	1	no	—
250	C₂×C₄⋊C₄	—	43	1	no	—
251	C₂×C₄⋊C₄	—	86	1	no	—
252	C₂×C₄⋊C₄	—	86	1	no	—
253	C₂×C₄⋊C₄	—	86	1	no	—
254	C₂×C₄⋊C₄	—	86	1	no	—
255	C₂×C₂²⋊C₄	—	86	1	no	—
256	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
257	#107	—	86	4	no	—
258	#106	—	86	4	no	—
259	C₄⋊C₈	—	2	1	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	43	2	no	—
262	C₄⋊C₈	—	86	1	no	—
263	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
264	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
265	#103	—	86	4	no	—
266	#103	—	86	4	no	—
267	#102	—	86	4	no	—
268	#102	—	86	4	no	—
269	C₂²⋊C₈	—	86	4	no	—
270	C₄×C₈	D₁₇₂	1	2	yes	—
271	C₄×C₈	C₂²×D₄₃	1	2	yes	—
272	C₂⁴	—	86	1	no	—
273	C₂×Q₈	—	86	2	no	—
274	C₂×Q₈	—	86	2	no	—
275	C₂×Q₈	—	344	2	no	—
276	C₂×Q₈	—	172	2	no	—
277	C₂×Q₈	—	172	2	no	—
278	C₂×Q₈	—	172	1	no	—
279	C₂×Q₈	—	172	1	no	—
280	C₂×D₄	—	43	2	no	—
281	C₂×D₄	—	86	1	no	—
282	C₂×D₄	—	86	2	no	—
283	C₂×D₄	—	86	2	no	—
284	C₂²×C₄	—	86	1	no	—
285	C₂²×C₄	—	86	2	no	—
286	C₂²×C₄	—	86	2	no	—
287	C₂²×C₄	—	172	1	no	—
288	C₂²×C₄	—	172	1	no	—
289	C₂²×C₄	—	172	2	no	—
290	C₂²×C₄	—	172	2	no	—
291	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
292	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
293	C₂²×C₄	#10621902	1	1	yes	characteristic
294	M₁₆	—	172	8	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	4	2	no	—
300	C₂×C₈	—	4	2	no	—
301	C₂×C₈	—	86	2	no	—
302	C₂×C₈	—	172	2	no	—
303	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
304	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
305	C₄⋊C₄	—	43	2	no	—
306	C₄⋊C₄	—	86	1	no	—
307	C₄⋊C₄	—	86	2	no	—
308	C₄⋊C₄	—	86	2	no	—
309	C₄⋊C₄	—	86	2	no	—
310	C₄⋊C₄	—	86	2	no	—
311	C₄⋊C₄	—	172	1	no	—
312	C₄⋊C₄	—	172	1	no	—
313	C₄⋊C₄	—	172	1	no	—
314	C₄⋊C₄	—	172	1	no	—
315	C₂²⋊C₄	—	86	2	no	—
316	C₂²⋊C₄	—	86	2	no	—
317	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
318	C₄²	D₄×D₄₃	1	1	yes	characteristic
319	C₄²	#10621902	1	1	yes	characteristic
320	C₄²	C₂³×D₄₃	1	1	yes	characteristic
321	C₂³	—	86	2	no	—
322	C₂³	—	172	2	no	—
323	C₂³	—	172	2	no	—
324	C₂³	—	172	2	no	—
325	C₂³	#13063300	1	1	yes	characteristic
326	Q₈	—	344	2	no	—
327	Q₈	—	344	2	no	—
328	Q₈	—	344	2	no	—
329	Q₈	—	172	2	no	—
330	Q₈	—	172	2	no	—
331	D₄	—	86	2	no	—
332	D₄	—	86	2	no	—
333	D₄	—	172	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	86	2	no	—
336	C₂×C₄	—	344	2	no	—
337	C₂×C₄	—	344	2	no	—
338	C₂×C₄	—	172	2	no	—
339	C₂×C₄	—	172	2	no	—
340	C₂×C₄	—	172	2	no	—
341	C₂×C₄	—	172	4	no	—
342	C₂×C₄	—	172	2	no	—
343	C₂×C₄	—	172	2	no	—
344	C₂×C₄	#13063300	1	1	yes	characteristic
345	C₂×C₄	#13063300	1	1	yes	characteristic
346	C₂×C₄	#13063300	1	1	yes	characteristic
347	C₂×C₄	#13063347	1	2	yes	—
348	C₂×C₄	#13063353	1	2	yes	—
349	C₂×C₄	#13063354	1	2	yes	—
350	C₂×C₄	#13063354	1	2	yes	—
351	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
352	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
353	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
354	C₈	—	2	2	no	—
355	C₈	—	2	2	no	—
356	C₈	—	4	2	no	—
357	C₈	—	4	2	no	—
358	C₈	—	172	2	no	—
359	C₂²	—	344	4	no	—
360	C₂²	—	172	2	no	—
361	C₂²	—	172	2	no	—
362	C₂²	—	172	2	no	—
363	C₂²	#14583115	1	2	yes	—
364	C₂²	#14583158	1	2	yes	—
365	C₂²	#14583163	1	2	yes	—
366	C₂²	C₂×#13063300	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	344	2	no	—
370	C₄	—	344	2	no	—
371	C₄	—	172	2	no	—
372	C₄	#14583115	1	2	yes	—
373	C₄	#14583158	1	2	yes	—
374	C₄	C₂×#13063347	1	1	yes	characteristic
375	C₄	C₂×#13063353	1	1	yes	characteristic
376	C₄	C₂×#13063354	1	1	yes	characteristic
377	C₄	C₂×#13063354	1	1	yes	characteristic
378	C₂	—	344	2	no	—
379	C₂	#19134171	1	4	yes	direct product factor
380	C₂	C₂×#14583115	1	1	yes	characteristic
381	C₂	C₂×#14583158	1	1	yes	characteristic
382	C₂	C₂×#14583163	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063300
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#334	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂²×D₁₇₂	D₄
hypercenter	C₂×#334	D₄₃
hypocenter	C₄₃	C₂×#1703
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#14583250	C₂	semidirect
5. #19134171	C₂	direct (N × H)
6. C₂×#14582647	C₂	semidirect
7. C₂×#14582644	C₂	semidirect
8. C₂×C₄₃×#334	C₂	semidirect
14. #14583250	C₂²	semidirect
29. #14582647	C₂²	semidirect
30. #14582644	C₂²	semidirect
34. C₄₃×#334	C₂²	semidirect
192. C₂×#334	D₄₃	semidirect
195. C₂×C₄₃	#1703	semidirect
223. #334	C₂×D₄₃	semidirect
226. C₄₃	C₂×#1703	semidirect
379. C₂	#19134171	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#14583250	2
C₂×#14583190	2
C₂×#14583180	2
C₂×#14582978	2
#19134171	2
C₂×#14582647	2
C₂×#14582644	2
C₂×C₄₃×#334	2
C₂×#1703	43

Subgroups — C2×#19134171 #25079171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19134171 #25079171